Singular Sums of Squares of Degenerate Vector Fields

In [7], J. J. Kohn proved C∞ hypoellipticity with loss of k − 1 derivatives in Sobolev norms (and at least that loss in L∞) for the highly non-subelliptic singular sum of squares Pk = LL + L|z| L = −L ∗ L − (zL)zL with L = ∂ ∂z + iz ∂ ∂t . In this paper, we prove hypoellipticity with loss of k−1 m derivatives in Sobolev norms for the operator (0.1) P F m,k = L F mL F m + L F m |z| Lm with L F m = ∂ ∂z + iFz ∂ ∂t , with F (z, z) such that (0.2) Fzz = |z| g, g(0) > 0, so that Fz = z|z| h whose prototype, when mF (z, z) = |z|2m, is (0.3) Pm,k = LmLm + Lm |z| Lm, Lm = ∂ ∂z + iz|z| ∂ ∂t , for which the underlying manifold is of finite type. We give two proofs: the first using a fairly rapid derivation of an a priori estimate analogous to that used by Kohn in [7]: (0.4) ‖φu‖0 ≤ C‖φ̃P F m,kv‖ k−1 m + C‖u‖−∞ (for all u ∈ C 0 with φ̃ ≡ 1 near supp φ), after deriving this estimate in the first part of the paper; the second uses the far more rapidly derived estimate of [12] and [5] (where analytic hypoellipticity for Pk and Pm,k are also proved): ∀v ∈ C 0 of small support, (0.5) ‖v‖ − k−1 2m + ‖Lv‖0 + ‖z Lv‖0 ≤ C|(P F m,kv, v)L2 |+ C‖v‖ 2 −N . We also prove, along the way, analytic hypoellipticity for P F m,k . For (0.6) F (z, z) = f(|z|), we show that these estimates are optimal.